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Physics as a Generalized Number Theory

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Year 2013



Riemann Hypothesis and quasicrystals

Freeman Dyson has represented a highly interesting speculation related to Riemann hypothesis and 1-dimensional quasicrystals (QCs). He discusses QCs and Riemann hypothesis briefly in his Einstein lecture.

Dyson begins from the defining property of QC as discrete set of points of Euclidian space for which the spectrum of wave vectors associated with the Fourier transform is also discrete. What this says is that quasicrystal as also ordinary crystal creates discrete diffraction spectrum. This presumably holds true also in higher dimensions than D=1 although Dyson considers mostly D=1 case. Thus QC and its dual would correspond to discrete points sets. I will consider the consequences in TGD framework below.

Dyson considers first QCs at general level. Dyson claims that QCs are possible only in dimensions D=1,2,3. I do not know whether this is really the case. In dimension D=3 the known QCs have icosahedral symmetry and there are only very few of them. In 2-D case (Penrose tilings) there is n-fold symmetry, roughly one kind of QC associated with any regular polygon. Penrose tilings correspond to n=5. In 1-D case there is no point group (subgroup of rotation group) and this explains why the number of QCs is infinite. For instance, so called PV numbers identified as algebraic integers, which are roots of any polynomial with integer coefficients such that all other roots have modulus smaller than unity. 1-D QCs is at least as rich a structure as PV numbers and probably much richer.

Dyson suggests that Riemann hypothesis and its generalisations might be proved by studying 1-D quasi-crystals.

  1. If Riemann Hypothesis is true, the spectrum for the Fourier transform of the distribution of zeros of Riemann zeta is discrete. The calculations of Andrew Odlycko indeed demonstrate this numerically, which is of course not a proof. From Dyson's explanation I understand that it consists of sums of integer multiples nlog(p) of logarithms of primes meaning that the non-vanishing Fourier components are apart from overall delta function (number of zeros) proportional to

    F(n)= ∑sk n-iskD(isk) , sk=1/2+iyk ,

    where sk are zeros of Zeta. ζD could be called the dual of zeta with summation over integers replaced with summation over zeros. For other "energies" than E=log(n) the Fourier transform would vanish. One can say that the zeros of Riemann Zeta and primes (or p-adic "energy" spectrum) are dual. Dyson conjectures that each generalized zeta function (or rather, L-function) corresponds to one particular 1-D QC and that Riemann zeta corresponds to one very special 1-D QC.

There are also intriguing connections with TGD, which inspire quaternionic generalization of Riemann Zeta and Riemann hypothesis.
  1. What is interesting that the same "energy" spectrum (logarithms of positive integers) appears in an arithmetic quantum field theory assignable to what I call infinite primes. An infinite hierarchy of second quantizations of ordinary arithmetic QFT is involved. A the lowest level the Fourier transform of the spectrum of the arithmetic QFT would consist of zeros of zeta rotated by π/2! The algebraic extensions of rationals and the algebraic integers associated with them define an infinite series of infinite primes and also generalized zeta functions obtained by the generalization of the sum formula. This would suggest a very deep connection with zeta functions, quantum physics, and quasicrystals. These zeta functions could correspond to 1-D QCs.
  2. The definition of p-adic manifold (in TGD framework) forces a discretisation of M4× CP2 having interpretation in terms of finite measurement resolution. This discretization induces also dicretization of space-time surfaces by induction of manifold structure. The discretisation of M4 (or E3) is achieved by crystal lattices, by QCs, and perhaps also by more general discrete structures. Could lattices and QCs be forced by the condition that the lattice like structures defines a discrete distributions with discrete spectrum? But why this?
  3. There is also another problem. Integration is a problematic notion in p-adic context and it has turned out that discretization is unavoidable and also natural in finite measurement resolution. The inverse of the Fourier transform however involves integration unless the spectrum of the Fourier transform is discrete so that in both E3 and corresponding momentum space integration reduces to a summation. This would be achieved if discretisation is by lattice or QC so that one would obtain the desired constraint on discretizations. Thus Riemann hypothesis has excellent mathematical motivations to be true in TGD Universe! Freeman Dyson has represented a highly interesting speculation related to Riemann hypothesis and 1-dimensional quasicrystals (QCs). He discusses QCs and Riemann hypothesis briefly in his Einstein lecture.

    Dyson begins from the defining property of QC as discrete set of points of Euclidian space for which the spectrum of wave vectors associated with the Fourier transform is also discrete. What this says is that quasicrystal as also ordinary crystal creates discrete diffraction spectrum. This presumably holds true also in higher dimensions than D=1 although Dyson considers mostly D=1 case. Thus QC and its dual would correspond to discrete points sets. I will consider the consequences in TGD framework below.

    Dyson considers first QCs at general level. Dyson claims that QCs are possible only in dimensions D=1,2,3. I do not know whether this is really the case. In dimension D=3 the known QCs have icosahedral symmetry and there are only very few of them. In 2-D case (Penrose tilings) there is n-fold symmetry, roughly one kind of QC associated with any regular polygon. Penrose tilings correspond to n=5. In 1-D case there is no point group (subgroup of rotation group) and this explains why the number of QCs is infinite. For instance, so called PV numbers identified as algebraic integers, which are roots of any polynomial with integer coefficients such that all other roots have modulus smaller than unity. 1-D QCs is at least as rich a structure as PV numbers and probably much richer.

    Dyson suggests that Riemann hypothesis and its generalisations might be proved by studying 1-D quasi-crystals.

    1. If Riemann Hypothesis is true, the spectrum for the Fourier transform of the distribution of zeros of Riemann zeta is discrete. The calculations of Andrew Odlycko indeed demonstrate this numerically, which is of course not a proof. From Dyson's explanation I understand that it consists of sums of integer multiples nlog(p) of logarithms of primes meaning that the non-vanishing Fourier components are apart from overall delta function (number of zeros) proportional to

      F(n)= ∑sk n-iskD(isk) , sk=1/2+iyk ,

      where sk are zeros of Zeta. ζD could be called the dual of zeta with summation over integers replaced with summation over zeros. For other "energies" than E=log(n) the Fourier transform would vanish. One can say that the zeros of Riemann Zeta and primes (or p-adic "energy" spectrum) are dual. Dyson conjectures that each generalized zeta function (or rather, L-function) corresponds to one particular 1-D QC and that Riemann zeta corresponds to one very special 1-D QC.

    There are also intriguing connections with TGD, which inspire quaternionic generalization of Riemann Zeta and Riemann hypothesis.
    1. What is interesting that the same "energy" spectrum (logarithms of positive integers) appears in an arithmetic quantum field theory assignable to what I call infinite primes. An infinite hierarchy of second quantizations of ordinary arithmetic QFT is involved. A the lowest level the Fourier transform of the spectrum of the arithmetic QFT would consist of zeros of zeta rotated by π/2! The algebraic extensions of rationals and the algebraic integers associated with them define an infinite series of infinite primes and also generalized zeta functions obtained by the generalization of the sum formula. This would suggest a very deep connection with zeta functions, quantum physics, and quasicrystals. These zeta functions could correspond to 1-D QCs.
    2. The definition of p-adic manifold (in TGD framework) forces a discretisation of M4× CP2 having interpretation in terms of finite measurement resolution. This discretization induces also dicretization of space-time surfaces by induction of manifold structure. The discretisation of M4 (or E3) is achieved by crystal lattices, by QCs, and perhaps also by more general discrete structures. Could lattices and QCs be forced by the condition that the lattice like structures defines a discrete distributions with discrete spectrum? But why this?
    3. There is also another problem. Integration is a problematic notion in p-adic context and it has turned out that discretization is unavoidable and also natural in finite measurement resolution. The inverse of the Fourier transform however involves integration unless the spectrum of the Fourier transform is discrete so that in both E3 and corresponding momentum space integration reduces to a summation. This would be achieved if discretisation is by lattice or QC so that one would obtain the desired constraint on discretizations. Thus Riemann hypothesis has excellent mathematical motivations to be true in TGD Universe!
    4. What could be the counterpart of Riemann Zeta in the quaternionic case? Quaternionic analog of Zeta suggests itself: formally one can define quaternionic zeta using the same formula as for Riemann zeta.
      1. Rieman zeta characterizes ordinary integers and s is in this case complex number, extension of reals by adding a imaginary unit. A naive generalization would be that quaternionic zeta characterizes Gaussian integers so that s in the sum ζ(s)=∑ n-s should be replaced with quaternion and n by Gaussian integer. In octonionic zeta s should be replaced with octonion and n with a quaternionic integer. The sum is well-defined despite the non-commutativity of quaternions (non-associativity of octonions) if the powers n-s are well-defined. Also the analytic continuation to entire quaternion/octonion plane should make sense and could be performed in a step wise manner by starting from real axis for s, extended to complex plane and then to quaternionic plane.
      2. Could the zeros sk of quaternionic zeta ζH(s) reside at the 3-D hyper-plane Re(q)=1/2, where Re(q) corresponds to E4 time coordinate (one must also be able to continue to M4)? Could the duals of zeros in turn correspond to logarithms ilog(n), n Gaussian integer. The Fourier transform of the 3-D distribution defined by the zeros would in turn be proportional to the dual of ζD,H(isk) of ζH. Same applies to the octonionic zeta.
      3. The assumption that n is ordinary integer in ζH would trivialize the situation. One obtains the distribution of zeros of ordinary Riemann zeta at each line s= 1/2+ yI, I any quaternionic unit and the loci of zeros would correspond to entire 2-spheres. The Fourier spectrum would not be discrete since only the magnitudes of the magnitudes of the quaternionic imaginary parts of "momenta" would be imaginary parts of zeros of Riemann zeta but the direction of momentum would be free. One would not avoid integration in the definition of inverse Fourier transform although the integrand would be constant in angular degrees of freedom.

    For background see the chapter Riemann Hypothesis and Physics.



p-Adic symmetries

The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the third one and devoted to p-adic symmetries.

A further objection relates to symmetries. It has become already clear that discrete subgroups of Lie-groups of symmetries cannot be realized p-adically without introducing algebraic extensions of p-adics making it possible to represent the p-adic counterparts of real group elements. Therefore symmetry breaking is unavoidable in p-adic context: one can speak only about realization of discrete sub-groups for the direct generalizations of real symmetry groups. The interpretation for the symmetry breaking is in terms of discretization serving as a correlate for finite measurement resolution reflecting itself also at the level of symmetries.

This observation has led to TGD inspired proposal for the realization of the p-adic counterparts symmetric spaces resembling the construction of P1(K) in many respects but also differing from it.

  1. For TGD option one considers a discrete subgroup G0 of the isometry group G making sense both in real context and for extension of p-adic numbers. One combines G0 with a p-adic counterpart of Lie group Gp obtained by exponentiating the Lie algebra by using p-adic parameters ti in the exponentiation exp(tiTi).
  2. One obtains actually an inclusion hierarchy of p-adic Lie groups. The levels of the hierarchy are labelled by the maximum p-adic norms |ti|p= p-ni, ni ≥ 1 and in the special case ni=n - strongly suggested by group invariance - one can write Gp,1 ⊃ Gp,2 ⊃ ...Gp,n .... Gp,i defines the p-adic counterpart of the continuous group which gets the smaller the larger the value of n is. The discrete group cannot be obtained as a p-adic exponential (although it can be obtained as real exponential), and one can say that group decomposes to a union of disconnected parts corresponding to the products of discrete group elements with Gp,n.

    This decomposition to totally uncorrelated disjoint parts is of course worrying from the point of view of algebraic continuation. The construction of p-adic manifolds by using canonical identification to define coordinate charts as real ones allows a correspondence between p-adic and real groups and also allows to glue together the images of the disjoint regions at real side: this induces gluing at p-adic side. The procedure will be discussed later in more detail.

  3. There is a little technicality is needed. The usual Lie-algebra exponential in the matrix representation contains an imaginary unit. For p mod 4 =3 this imaginary unit can be introduced as a unit in the algebraic extension. For p mod 4 =1 it can be realized as an algebraic number. It however seems that imaginary unit or its p-adic analog should belong to an algebraic extension of p-adic numbers. The group parameters for algebraic extension of p-adic numbers belong to the algebraic extension. If the algebraic extension contains non-trivial roots of unity Um,n= exp(i2 π m/n), the differences Um,n-U*m,n are proportional to imaginary unit as real numbers and one can replace imaginary unit in the exponential with Um,n-U*m,n. In real context this means only a rescaling of the Lie algebra generator and Planck constant by a factor (2sin(2 π m/n))-1. A natural imaginary unit is defined in terms of U1,pn.
  4. This construction is expected to generalize to the case of coset spaces and give rise to a coset space G/H identified as the union of discrete coset spaces associated with the elements of the coset G0/H0 making sense also in the real context. These are obtained by multiplying the element of G0/H0 by the p-adic factor space Gp,n/Hp,n.

One has two hierarchies corresponding to the hierarchy of discrete subgroups of G0 requiring each some minimal algebraic extension of p-adic numbers and to the hierarchy of Gp:s defined by the powers of p. These two hierarchies can be assigned to angles (actually phases coming as roots of unity) and p-adic length scales in the space of group parameters.

The Lie algebra of the rotation group spanned by the generators Lx,Ly,Lz provides a good example of the situation and leads to the question whether the hierarchy of Planck constants kenociteallb/Planck could be understood p-adically.

  1. Ordinary commutation relations are [Lx,Ly]= i hbar Lz. For the hierarchy of Lie groups it is convenient to extend the algebra by introducing the generators Lin)= pnLi and one obtains [Lxm),Lyn)]= i hbar Lzm+n). This resembles the commutation relations of Kac-Moody algebra structurally.
  2. For the generators of Lie-algebra generated by Lim) one has [Lxm),Lym)]= ipm hbar Lzm). One can say that Planck constant is scaled from hbar to pm hbar. Could the effective hierarchy of Planck constants assigned to the multi-furcations of space-time sheets correspond in p-adic context to this hierarchy of Lie-algebras?
  3. The values of the Planck constants would come as powers of primes: the hypothesis has been that they comes as positive integers. The integer n defining the number of sheets for n-furcation would come as powers n=pm. The connection between p-adic length scale hierarchy and hierarchy of Planck constants has been conjectured already earlier but the recent conjecture is the most natural one found hitherto. Of course, the question whether the number sheets of furcation correlates with the power of p characterizing "small" continuous symmetries remains an open question. Note that also n-adic and even q=m/n-adic topology is possible with norms given by powers of integer or rational. Number field is however obtained only for primes. This suggests that if also integer - and perhaps even rational valued scales are allowed for causal diamonds, they correspond to effective n-adic or q-adic topologies and that powers of p are favored.

The difficult questions concern again integration. The integrals reduce to sums over the discrete subgroup of G multiplied with an integral over the p-adic variant Gp,n of the continuous Lie group. The first integral - that is summation - is number theoretically universal. The latter integral is the problematic one.
  1. The easy way to solve the problem is to interpret the hierarchy of continuous p-adic Lie groups Gp,n as analogs of gauge groups. But if the wave functions are invariant under Gp,n, what is the situation with respect to Gp,m for m<n? Infinitesimally one obtains that the commutator algebras [Gp,k,Gp,l] ⊂ Gp,k+l must annihilate the functions for k+l ≥ n. Does also Gp,m, m<n annihilate the functions for as a direct calculation demonstrates in the real case. If this is the case also p-adically the hierarchy of groups Gp,n would have no physical implications. This would be disappointing.
  2. One must however be very cautious here. Lie algebra consists of first order differential operators and in p-adic context the functions annihilated by these operators are pseudo-constants. It could be that the wave functions annihilated by Gp,n are pseudo-constants depending on finite number of pinary digits only so that one can imagine of defining an integral as a sum. In the recent case the digits would naturally correspond to powers pm, m<n. The presence of these functions could be purely p-adic phenomenon having no real counterpart and emerge when one leaves the intersections of real and p-adic worlds. This would be just the non-determinism of imagination assigned to p-adic physics in TGD inspired theory of consciousness.

Is there any hope that one could define harmonic analysis in Gp,n in a number theoretically universal manner? Could one think of identifying discrete subgroups of Gp,n allowing also an interpretation as real groups?
  1. Exponentiation implies that in matrix representation the elements of Gp,n are of form g= Id+ png1: here Id represents real unit matrix. For compact groups like SU(2) or CP2 the group elements in real context are bounded above by unity so that this kind of sub-groups do not exist as real groups. For non-compact groups like SL(2,C) and T4 this kind of subgroups make sense also in real context.
  2. Zero energy ontology suggests that discrete but infinite sub-groups Γ of SL(2,C) satisfying certain additional conditions define hyperbolic spaces as factor spaces H3/ Γ (H3 is hyperboloid of M4 lightcone). These spaces have constant sectional curvature and very many 3-manifolds allow a hyperbolic metric with hyperbolic volume defining a topological invariant. The moduli space of CDs contains the groups Γ defining lattices of H3 replacing it in finite measurement resolution. One could imagine hierarchies of wave functions restricted to these subgroups or H3 lattices associated with them. These wave functions would have the same form in both real and p-adic context so that number theoretical universality would make sense and one could perhaps define the inner products in terms of "integrals" reducing to sums.
  3. The inclusion hierarchy Gp,n ⊃ Gp,n+1 would in the case of SL(2,C) have interpretation in terms of finite measurement resolution for four-momentum. If Gp,n annihilate the physical states or creates zero norm states, this inclusion hierarchy corresponds to increasing IR cutoff (note that short length scale in p-adic sense corresponds to long scale in real sense!). The hierarchy of groups Gp,n makes sense also in the case of translation group T4 and also now the interpretation in terms of increasing IR cutoff makes sense. This picture would provide a group theoretic realization for with the vision that p-adic length scale hierarchies correspond to hierarchies of length scale measurement resolutions in M4 degrees of freedom.

Canonical identification and the definition of p-adic counterparts of Lie groups

For Lie groups for which matrix elements satisfy algebraic equations, algebraic subgroups with rational matrix elements could regarded as belonging to the intersection of real and p-adic worlds, and algebraic continuation by replacing rationals by reals or p-adics defines the real and p-adic counterparts of these algebraic groups. The challenge is to construct the canonical identification map between these groups: this map would identify the common rationals and possible common algebraic points on both sides and could be seen also a projection induced by finite measurement resolution.

A proposal for a construction of the p-adic variants of Lie groups was discussed in previous section. It was found that the p-adic variant of Lie group decomposes to a union of disjoint sets defined by a discrete subgroup G0 multiplied by the p-adic counterpart Gp,n of the continuous Lie group G. The representability of the discrete group requires an algebraic extension of p-adic numbers. The disturbing feature of the construction is that the p-adic cosets are disjoint. Canonical identification Ik,l suggests a natural solution to the problem. The following is a rough sketch leaving a lot of details open.

  1. Discrete p-adic subgroup G0 corresponds as such to its real counterpart represented by matrices in algebraic extension of rationals. Gp,n can be coordinatized separately by Lie algebra parameters for each element of G0 and canonical identification maps each Gp,n to a subset of real G. These subsets intersect and the chart-to-chart identification maps between Lie algebra coordinates associated with different elements of G0 are defined by these intersections. This correspondence induces the correspondence in p-adic context by the inverse of canonical identification.
  2. One should map the p-adic exponentials of Lie-group elements of Gp,n to their real counterparts by some form of canonical identification.

    1. Consider first the basic form I=I0, ∞ of canonical identification mapping all p-adics to their real counterparts and maps only the p-adic integers 0 ≤ k<p to themselves.

      The gluing maps between groups Gp,n associated with elements gm and gn of G0 would be defined by the condition gm I(exp(itaTa)= gn I(exp(ivaTa). Here ta and va are Lie-algebra coordinates for the groups at gm and gn. The delicacies related to the identification of p-adic analog of imaginary unit have been discussed in the previous section. It is important that Lie-algebra coordinates belong to the algebraic extension of p-adic numbers containing also the roots of unity needed to represent gn. This condition allows to solve va in terms of ta and va= va(tb) defines the chart map relating the two coordinate patches on the real side. The inverse of the canonical identification in turn defines the p-adic variant of the chart map in p-adic context. For I this map is not p-adically analytic as one might have guessed.

    2. The use of Ik,n instead of I gives hopes about analytic chart-to chart maps on both sides. One must however restrict Ik,n to a subset of rational points (or generalized points in algebraic extension with generalized rational defined as ratio of generalized integers in the extension). Canonical identification respects group multiplication only if the integers defining the rationals m/n appearing in the matrix elements of group representation are below the cutoff pk. The points satisfying this condition do not in general form a rational subgroup. The real images of rational points however generate a rational sub-group of the full Lie-group having a manifold completion to the real Lie-group.

      One can define the real chart-to chart maps between the real images of Gp,k at different points of G0 using Ik,l(exp(ivaTa)= gn-1gm × Ik,l(exp(itaTa). When real charts intersect, this correspondence should allow solutions va,tb belonging to the algebraic extension and satisfying the cutoff condition. If the rational point at the other side does not correspond to a rational point it might be possible to perform pinary cutoff at the other side.

      Real chart-to-chart maps induce via common rational points discrete p-adic chart-to-chart maps between Gp,k. This discrete correspondence should allow extension to a unique chart-to-chart map the p-adic side. The idea about algebraic continuation suggests that an analytic form for real chart-to-chart maps using rational functions makes sense also in the p-adic context.

  3. p-Adic Lie-groups Gp,k for an inclusion hierarchy with size characterized by p-k. For large values of k the canonical image of Gp,k for given point of G0 can therefore intersect its copies only for a small number of neighboring points in G0, whose size correlates with the size of the algebraic extension. If the algebraic extension has small dimension or if k becomes large for a given algebraic extension, the number of intersection points can vanish. Therefore it seems that in the situations, where chart-to-chart maps are possible, the power pk and the dimension of algebraic extension must correlate. Very roughly, the order of magnitude for the minimum distance between elements of G0 cannot be larger than p-k+1. The interesting outcome is that the dimension of algebraic extension would correlate with the pinary cutoff analogous to the IR cutoff defining measurement resolution for four-momenta.

For details see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.



Could canonical identification make possible definition of integrals in p-adic context?

The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the second one and devoted to p-adic integration.

The notion of p-adic manifold using using real chart maps instead of p-adic ones allows an attractive approach also to p-adic integration and to the problem of defining p-adic version of differential forms and their integrals.

  1. If one accepts the simplest form of canonical identification I(x): ∑n xnpn → ∑ xnp-n, the image of the p-adic surface is continuous but not differentiable and only integers n<p are mapped to themselvs. One can define integrals of real functions along images of the p-adically analytic curves and define the values of their p-adic counterparts as their algebraic continuation when it exists.

    In TGD framework this does not however work. If one wants to define induced quantities - such as metric and K ähler form - on the real side one encounters a problem since the image surface is not smooth and the presence of edges implies that these quantities containing derivatives of imbedding space coordinates possess delta function singularities. These singularities could be even dense in the integration region so that one would have no-where differentiable continuous functions and the real integrals would reduce to a sum which do not make sense.

  2. In TGD framework finite measurement resolution realized in terms of pinary cutoff however saves the situation. The canonical identification Ik,l(m/n) = Ik,l(m)/Ikl(n) maps rationals to themselves for m<pk,n<pk. The second pinary cutoff m<pl,n<pl, l>k implies that the chart map takes a discrete subset of p-adic rationals to a discrete set of real rationals. The completion of the discrete image of p-adic preferred extremal under Ik,l to a real preferred extremal is very natural. This preferred extremal can be said to be unique apart from a finite measurement resolution represented by the pinary cutoffs k and l. All induced quantities are well defined on both sides.

    p-Adic integrals can be defined as pullbacks of real integrals by algebraic continuation when this is possible. The inverse image of the real integration region in canonical identification defines the p-adic integration region.

  3. The integrals of p-adic differential forms can be defined as pullbacks of the real integrals. The integrals of closed forms, which are typically integers, would be the same integers but interpreted as p-adic integers.

It is interesting to study the algebraic continuation of K ähler action from real sector to p-adic sectors.
  1. K ähler action for both Euclidian and Minkowskian regions reduces to the algebraic continuation of the integral of Chern-Simons-K ähler form over preferred 3-surfaces. The contributions from Euclidian and Minkowskian regions reduce to integrals of Chern-Simons form over 3-surfaces. I have somewhere considered the possibility that the 3-surfaces for Minkowskian and Euclidian contribution might be identical: this cannot be the case since the space-like 3-surfaces at the boundaries of CD for Minkowskian and Euclidian regions are disjoint.

    The contribution from Euclidian regions defines K ähler function of WCW and the contribution from Minkowskian regions giving imaginary exponential of K ähler action has interpretation as Morse function whose stationary points are expected to select special preferred extremals. One would expect that both functions have a continuous spectrum of values. In the case of K ähler function this is necessary since K ähler function defines the K ähler metric of WCW via its second derivatives in complex coordinates by the well-known formula. Note that by the above observation K ähler and Morse functions are not in general proportional to each other.

  2. The algebraic continuation of the exponent of K ähler function for a given p-adic prime is expected to require the proportionality to pn so that not all preferred extremals are expected to allow a continuation to a given p-adic number field. This kind of assumption has been indeed made in the case of deformations of CP2 type extremals in order to derive formula for the gravitational constant in terms of basic parameters of TGD but without real justification (see this).
  3. The condition that the action exponential in the Minkowskian regions is a genuine phase factor implies that it reduces to a root of unity (one must have an algebraic extension of p-adic numbers). Therefore the contribution to the imaginary exponent K ähler action from these regions for the p-adicizable preferred extremals should be of form 2 π (k+m/n).

    If all preferred real extremals allow p-adic counterpart, the value spectrum of the Morse function on the real side is discrete and could be forced by the preferred extremal property. If this were the case the stationary phase approximation around extrema of K ähler function on the real side would be replaced by sum with varying phase factors weighted by K ähler function.

    An alternative conclusion is that the algebraic continuation of K ähler action to any p-adic field is possible only for a subset of preferred extremals with a quantized spectrum of Morse function. One the real side stationary phase approximation would make sense. It however seems that the stationary phases must obey the above discussed quantization rule.

Also holomorphic forms allow algebraic continuation and one can require that also their integrals over cycles do so. An important example is provided by the holomorphic one-forms integrals over cycles of partonic 2-surface defining the Teichmueller parameters characterizing the conformal equivalence class of the partonic 2-surfaces as Riemann surface. The p-adic variants exist of these parameters exist if they allow an algebraic continuation to a p-adic number. The algebraic continuation from the real side to the p-adic side would be possible on for certain p-adic primes p if any: this would allow to assign p-adic prime or primes to a given real preferred extremal. This justifies the assumptions of p-adic mass calculations concerning the contribution of conformal modular degrees of freedom to mass squared (see this).

For details see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.



Could canonical identification allow construction of path connected topologies for p-adic manifolds?

The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the first one and devoted to the notion of p-adic manifold.

Total disconnectedness of p-adic numbers as the basic problem

The total dis-connectedness of p-adic topology and lacking correspondence with real manifolds could be seen as genuine problem in the purely formal construction of p-adic manifolds. Physical intuition suggests that path connected should be realized in some natural manner and that one should have a close connection with real topology which after all is the "lab topology".

In TGD framework one of the basic physical problems has been the connection between p-adic numbers and reals. Algebraic and topological approaches have been competing also here.

  1. Algebraic approach suggests the identification of reals and various p-adic numbers along common rationals but this correspondence is non-continuous. Above some resolution defined by power of p it must be replaced with a correspondence is continuous unless one uses pinary cutoff. Below this cutoff the pseudo-constants of p-adic differential equations would naturally relate to the identification of p-adics and reals along common rationals (plus common algebraics in the case of algebraic extensions).
  2. Topological approach relies on canonical identification and its variants mapping p-adic numbers to reals in a continuous manner. This correspondence is however problematic in the sense that does not commute with the basic symmetries as correspondence along common rationals would do for subgroups of the symmetries represented in terms of rational matrices. A further problematic aspect of canonical identification is that it does not commute with the field equations.
  3. The notion of finite measurement resolution allows to find a compromise between the symmetries and continuity (that is, algebra and topology). Canonical identification can be modified so that it maps rationals to themselves only up to some pinary digits but is still continuous in p-adic sense. Canonical identification could map only a skeleton formed by discrete point set - analogous to Bruhat-Tits building - from real to p-adic context and the preferred extremals on both sides would contain this skeleton.

Canonical identification combined with the identification of common rationals in finite pinary resolution suggests also a manner of replacing p-adic topology with a path connected one. This topology would be essentially real topology induced to p-adic context by canonical identification used to build real chart leafs.
  1. Canonical identification maps p-adic numbers ∑ xnpn to reals and is defined by the formula I(x) = ∑ xnp-n. I is a continuous map from p-adic numbers to reals. Its inverse is also continuous but two-valued for a finite number of pinary digits since the pinary expansion of real number is not unique (1=.999999.. is example of this in 10-adic case). For a real number with a finite number of pinary digits one can always choose the p-adic representative with a finite number of pinary digits.
  2. Canonical identification is used to map the predictions of p-adic mass calculations to map the p-adic value of the mass squared to its real counterpart. It makes also sense to map p-adic probabilities to their real counterparts by canonical identification. In TGD inspired theory of consciousness canonical identification is a good candidate for defining cognitive representations as representations mapping real preferred extremals to p-adic preferred extremals as also for the realization of intentional action as a quantum jump replacing p-adic preferred extremal representing intention with a real preferred extremal representing action. Could these cognitive representations and their inverses actually define real coordinate charts for the p-adic "mind stuff" and vice versa?
  3. Canonical identification has several variants. For instance, one can map p-adic rational number m/n regarded as a p-adic number to a real number I(m)/I(n). In this case canonical identification respects rationality but is ill-defined for p-adic irrationals. This is not a catastrophe if one has finite measurement resolution meaning that only rationals for which m<pl,n<pl are mapped to the reals (real rationals actually).

    One can also express p-adic number as expansion of powers fo pk and define canonical identification Ik as ∑ xnpkn → ∑ xnp-kn. Also the variant Ik,l(m/n)=Ik,l(m)/Ik,l(n) with l defining pinary cutoff for m and l makes sense. One can say that Ik,l(m/n) identifies p-adic and real numbers along common rationals for p-adic numbers with a pinary cutoff defined by k and maps them to rationals for pinary cutoff defined by l. Discrete subset of rational points on p-adic side is mapped to a discrete subset of rational points on real side by this hybrid of canonical identification and identification along common rationals. This form of canonical identification is the one needed in TGD framework.

  4. Canonical identification does not commute with rational symmetries unless one uses the map Ik,l(m/n)=Ik,l(m)/Ik,l(n) and also now only in finite resolutions defined by k. For the large p-adic primes associated with elementary particles this is not a practical problem (electron corresponds to M127=2127-1!) The generalization to algebraic extensions makes also sense. Canonical identification breaks general coordinate invariance unless one uses group theoretically preferred coordinates for M4 and CP2 and subset of these for the space-time region considered.

What is very remarkable is that canonical identification can be seen as a continuous generalization of the p-adic norm defined as Np(x) == Ik,l(x) having the highly desired Archimedean property. Ik,l is the most natural variant of canonical identification.
  1. Canonical identification for the various coordinates defines a chart map mapping regions of p-adic manifold to Rn+. That each coordinate is mapped to a norm Np(x) means that the real coordinates are always non-negative. If real spaces Rn+ would provide only chart maps, it is not necessary to require approximate commutativity with symmetries. Also Berkovich considers norms but for a space of formal power series assigned with the p-adic disk: in this case however the norms have extremely low information content.
  2. Ik,l(x) indeed defines the analog of Archimedean norm in the sense that one has Np(x+y) ≤ Np(x)+Np(y). This follows immediately from the fact that the sum of pinary digits can vanish modulo p. The triangle inequality holds true also for the rational variant of I. Np(x) is however not multiplicative: only a milder condition Np(pnx)=N(pn)N(x)=p-n N(x) holds true.
  3. Archimedean property gives excellent hopes that p-adic space provided with chart maps for the coordinates defined by canonical identification inherits real topology and its path connectedness. A hierarchy of topologies would be obtained as induced real topologies and characterized by various norms defined by Ik,l labelled by a finite measurement resolution. This would give a very close connection with physics.
  4. The mapping of p-adic manifolds to real manifolds would make the construction of p-adic topologies very concrete. For instance, one can map real preferred subset of rationalp oints of a real extremal to a p-adic one by the inverse of canonical identification by mapping the real points with finite number of pinary digits to p-adic points with a finite number of pinary digits. This does not of course guarantee that the p-adic preferred extremal is unique. One could however hope that p-adic preferred extrremals can be said to possess the invariants of corresponding real topologies in finite measurement resolution.
  5. The maps between different real charts would be induced by the p-adically analytic maps between the inverse images of these charts. At the real side the maps would be consistent with the p-adic maps only in the discretization below pinary cutoff.
  6. As already mentioned, one must restrict the p-adic points mapped to reals to rationals since Ik,l(m/n) is not well-defined for p-adic irrationals (having non-periodic pinary expansion: note however that one can consider also p-adic integers). For the restriction to finite rationals the chart image on real side would consist of rational points. The cutoff would mean that these rationals are not dense in the set of reals. Preferred extremal property could however allow to identify the chart leaf as a piece of preferred extremal containing the rational points in the measurement resolution use. This would realize the dream of mapping p-adic p-adic preferred extremals to real ones playing a key role in number theoretical universality.

To sum up, chart maps are constructed in two steps and works in both directions. For p-adic-to-real case a subset of rational points of the p-adic preferred extremal would be mapped using Ik,l to rational points of the real preferred extremal. Field equations for the preferred extremal would be then used to complete the resulting discrete skeleton to a full map leaf. Of course also algebraic extensions can and must be considered. This kind of completion performed in iterative manner has been also proposed assuming that space-time surfaces are quaternionic surfaces (tangent spaces are in well-defined sense quaterionic sub-space of octonionic space containing complex octonions as a preferred sub-space this).

What about p-adic coordinate charts for a real preferred extremal?

What is remarkable that one can also build p-adic coordinate charts about real preferred extremal using the inverse of the canonical identification assuming that finite rationals are mapped to finite rationals. There are actually good reasons to expect that coordinate charts make sense in both directions.

Algebraic continuation from real to p-adic context is one such reason. At the real side one can calculate the values of various integrals like K ähler action. This would favor p-adic regions as map leafs. One can require that K ähler action for Minkowskian and Euclidian regions (or their appropriate exponents) make sense p-adically and define the values of these functions for the p-adic preferred extremals by algebraic continuation. This could be very powerful criterion allowing to assign only very few p-adic primes to a given real space-time surface. This would also allow to define p-adic boundaries as images of real boundaries in finite measurement resolution. p-Adic path connectedness would be induced from real path-connectedness.

p-Adic rationals include also the ratios of integers, which are infinite as real integers so that the pinary expansion of the rational is not periodic asymptotically. In principle one could imagine of mapping also these to real numbers but the resulting skeleton might be too dense and might not allow to satisfy the preferred extremal property. Furthermore, the representation of a p-adic number as a ratio of this kind of integers is not unique and can be always tranformed to an infinite p-adic integer multiplied by a power of p . In the same manner real points which can be regarded as images of ratios of p-adic integers infinite as real integers could be mapped to p-adic ones but same problem is encountered also now.

In the intersection of real and p-adic worlds the correspondence is certainly unique and means that one interprets the equations defining the p-adic space-time surface as real equations. The number of rational points (with cutoff) for the p-adic preferred extremal becomes a measure for how unique the chart map in the general case can be. For instance, for 2-D surfaces the surfaces xn+yn=zn allow no nontrivial rational solutions for n>2 for finite real integers. This criterion does not distinguish between different p-adic primes and algebraic continuation is needed to make this distinction.

Chart maps for p-adic manifolds

The real map leafs must be mutually consistent so that there must be maps relating coordinates used in the overlapping regions of coordinate charts on both real and p-adic side. On p-adic side chart maps between real map leafs are naturally induced by identifying the canonical image points of identified p-adic points on the real side. For discrete chart maps Ik,l with finite pinary cutoffs one one must complete the real chart map to - say diffeomorphism. That this completion is not unique reflects the finite measurement resolution.

In TGD framework the situation is dramatically simpler. For sub-manifolds the manifold structure is induced from that of imbedding space and it is enough to construct the manifold structure M4 × CP2 in a given measurement resolution (k,l). Due to the isometries of the factors of the imbedding space, the chart maps in both real and p-adic case are known in preferred imbedding space coordinates. As already discussed, this allows to achieve an almost complete general coordinate invariance by using subset of imbedding space coordinates for the space-time surface. The breaking of GCI has interpretation in terms of presence of cognition and selection of quantization axes.

For instance, in the case of Riemann sphere S2 the holomorphism relating the complex coordinates in which rotations act as M öbius tranformations and rotations around -call it z-axis- act as phase multiplications - the coordinates z and w at Norther and Southern hemispheres are identified as w=1/z restricted to rational points at both side. For CP2 one has three poles instead of two but the situation is otherwise essentially the same.

For details see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.



What p-adic icosahedron could mean? And what about p-adic manifold?

I have been working for a couple of weeks with the problem of defining the notion of p-adic manifold: this is one of the key challenges of TGD. The existing proposals by mathematicians are rather complicated and it seems that something is lacking. To my opinion, to identify this something it is essential to make the question "What p-adic numbers are supposed to describe?". This question has not bothered either matheticians or theoretical physicists proposing purely formal p-adic counterparts for the scattering amplitudes.

Without any answer to this question there are simply quite too many alternatives to consider and one ends up to the garden of branching paths. The text below is this introduction to the article and chapter about the topics.

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This article was originally meant to be a summary of what I understand about the article "The p-Adic Icosahedron" in Notices of AMS. The original purpose was to summarize the basic ideas and discuss my own view about more technical aspects - in particular the generalization of Riemann sphere to p-adic context which is rather technical and leads to the notion of Bruhat Tits tree and Berkovich space. About Bruhat-Tits tree there is a nice web article titled p-Adic numbers and Bruhat-Tits tree describing also basics of p-adic numbers in a very concise form.

The notion of p-adic icosahedron leads to the challenge of constructing p-adic sphere, and more generally p-adic manifolds and this extended the intended scope of the article and led to consider the fundamental questions related to the construction of TGD.

Quite generally, there are two approaches to the construction of manifolds based on algebra resp. topology.

  1. In algebraic geometry manifolds - or rather, algebraic varieties - correspond to solutions of algebraic equations. Algebraic approach allows even a generalization of notions of real topology such as the notion of genus.
  2. Second approach relies on topology and works nicely in the real context. The basic building brick is n-ball. More complex manifolds are obtained by gluing n-balls together. Here inequalities enter the game. Since p-adic numbers are not well-ordered they do not make sense in purely p-adic context unless expressed using p-adic norm and thus for real numbers. The notion of boundary is also one of the problematic notions since in purely p-adic context there are no boundaries.

The attempt to construct p-adic manifolds by mimicking topological construction of real manifolds meets difficulties

The basic problem in the application of topological method to manifold construction is that p-adic disks are either disjoint or nested so that the standard construction of real manifolds using partially overlapping n-balls does not generalize to the p-adic context. The notions of Bruhat-Tits tree, building, and Berkovich disks and Berkovich space are represent attempts to overcome this problem. Berkovich disk is a generalization of the p-adic disk obtained by adding additional points so that the p-adic disk is a dense subset of it. Berkovich disk allows path connected topology which is not ultrametric. The generalization of this construction is used to construct p-adic manifolds using the modification of the topological construction in the real case. This construction provides also insights about p-adic integration.

The construction is highly technical and complex and pragmatic physicist could argue that it contains several un-natural features due to the forcing of the real picture to p-adic context. In particular, one must give up the p-adic topology whose ultra-metricity has a nice interpretation in the applications to both p-adic mass calculations and to consciousness theory.

I do not know whether the construction of Bruhat-Tits tree, which works for projective spaces but not for Qpn (!) is a special feature of projective spaces, whether Bruhat-Tits tree is enough so that no completion would be needed, and whether Bruhat-Tits tree can be deduced from Berkovich approach. What is remarkable that for M4× CP2 p-adic S2 and CP2 are projective spaces and allow Bruhat-Tits tree. This not true for the spheres associated with the light-cone boundary of D≠ 4-dimensional Minkowski spaces.

Two basic philosophies concerning the construction of p-adic manifolds

There exists two basic philosophies concerning the construction of p-adic manifolds: algebraic and topological approach. Also in TGD these approaches have been competing: algebraic approach relates real and p-adic space-time points by identifying common rationals. Finite pinary cutoff is however required to achieve continuity and has interpretation in terms of finite measurement resolution. Canonical identification maps p-adics to reals and vice versa in a continuous manner but is not consistent with field equations without pinary cutoff.

  1. One can try to generalize the theory of real manifolds to p-adic context. Since p-adic balls are either disjoint or nested, the usual constuction by gluing partially overlapping balls fails. This leads to the notion of Berkovich disk obtained as a completion of p-adic disk having path connected topology (non-ultrametric) and containing p-adic disk as a dense subset. This plus the complexity of the construction is heavy price to be paid for path-connectedness. A related notion is Bruhat-Tits tree defining kind of skeleton making p-adic manifold defining its boundary path connected. The notion makes sense for the p-adic counterparts of projective spaces, which suggests that p-adic projective spaces (S2 and CP2 in TGD framework) are physically very special.
  2. Second approach is algebraic and restricts the consideration to algebraic varieties for which also topological invariants have algebraic counterparts. This approach is very natural in TGD framework, where preferred extremals of Kähler action can be characterized purely algebraically - even in a manner independent of the action principle - so that they make sense also p-adically.

    At the level of WCW algebraic approach combined with symmetries works: the mere existence of Kähler geometry implies infinite-D group of isometries and fixes the geometry uniquely. One can say that infinite-D geometries are the final victory of Erlangen program. At space-time level it however seems that one must have correspondence between real and p-adic worlds since real topology is the "lab topology". Canonical identification should enter the construction.

Number theoretical universality and the construction of p-adic manifolds

Construction of p-adic counterparts of manifolds is also one of the basic challenges of TGD. Here the basic vision is that one must take a wider perspective. One must unify real and various p-adic physics to single coherent whole and to relate them. At the level of mathematics this requires fusion of real and p-adic number fields along common rationals and the notion of algebraic continuation between number fields becomes a basic tool.

The number theoretic approach is essentially algebraic and based on the gluing of reals and various p-adic number fields to a larger structure along rationals and also along common algebraic numbers. A strong motivation for the algebraic approach comes from the fact that preferred extremals are characterized by a generalization of the complex structure to 4-D case both in Euclidian and Minkowskian signature. This generalization is independent of the action principle. This allows a straightforward identification of the p-adic counterparts of preferred extremals. The algebraic extensions of p-adic numbers play a key role and make it possible to realize the symmetries in the same manner as they are realized in the construction of p-adic icosahedron.

The lack of well-ordering of p-adic numbers implies strong constraints on the formulation of number theoretical universality.

  1. The notion of set theoretic boundary does not make sense in purely p-adic context. Quite, generally everything involving inequalities can lead to problems in p-adic context unless one is able to define effective Archimedean topology in some natural manner. Canonical identifcation inducing real topology to p-adic context would allow to achieve this.
  2. The question arises about whether real topological invariants such as genus of partonic 2-surface make sense in the p-adic sector: for algebraic varieties this is the case. One would however like to have a more general definition and again Archimedean effective topology is suggestive.
  3. Integration poses problems in p-adic context and algebraic continuation from reals to p-adic number fields seems to be the only possible option making sense. The continuation is however not possible for all p-adic number fields for given surface. This has however a beautiful interpretation explaining why real space-time sheets (and elementary particles) are characterized by some p-adic prime or primes. The p-adic prime determining the mass scale of the elementary particle could be fixed number theoretically rather than by some dynamical principle formulated in real context (number theoretic anatomy of rational number does not depend smoothly on its real magnitude!). A more direct approach to integration could rely on canonical integration as a chart map allowing to define integral on the real side.
  4. Only those discrete subgroups of real symmetries, which correspond matrices with elements in algebraic extension of p-adic numbers can be realized so that a symmetry breaking to discrete subgroup consistent with the notion of finite measurement resolution and quantum measurement theory takes place. p-Adic symmetry groups can be identified as unions of elements of discrete subgroup of the symmetry group (making sense also in real context) multiplied by a p-adic variant of the continuous Lie group. These genuinely p-adic Lie groups are labelled by powers of p telling the maximum norm of the Lie-algebra parameter. Remarkably, effective values of Planck constant come as powers of p. Whether this interpretation for the hierarchy of effective Planck constants is consistent with the interpretation in terms of n-furcations of space-time sheet remains an open question.

How to achieve path connectedness?

The basic problem in the construction of p-adic manifolds is the total disconnectedness of the p-adic topology implied by ultrametricity. This leads also to problems with the notion of p-adic integration. Physically it seems clear that the notion of path connectedness should have some physical counterpart.

The notion of open set makes possib le path connectedness possible in the real context. In p-adic context Bruhat-Tits tree and Berkovich disk are introduced to achieve the same goal. One can of course ask whether Berkovich space could allow to achieve a more rigorous formulation for the p-adic counterparts of CP2, of partonic 2-surfaces, their light-like orbits, preferred extremals of Kähler action, and even the "world of classical worlds" (WCW). To me this construction does not look promising in TGD framework but I could be wrong.

TGD suggests two alternative approaches to the problem of path connectedness. They should be equivalent.

p-Adic manifold concept based on canonical identification

The TGD inspired solution to the construction of path connectd p-adic topology is based on the notion of canonical identification mapping reals to p-adics and vice versa in a continuous manner.

  1. Canonical identification is used to map the predictions of p-adic mass calculations to map the p-adic value of the mass squared to its real counterpart. It makes also sense to map p-adic probabilities to their real counterparts by canonical identification. In TGD inspired theory of consciousness canonical identification is a good candidate for defining cognitive representations as representations mapping real preferred extremals to p-adic preferred extremals as also for the realization of intentional action as a quantum jump replacing p-adic preferred extremal representing intention with a real preferred extremal representing action. Could these cognitive representations and their inverses actually define real coordinate charts for the p-adic "mind stuff" and vice versa?
  2. The trivial but striking observation was that it satisfies triangle inequality and thus defines an Archimedean norm allowing to induce real topology to p-adic context. Canonical identification with finite measurement resolution defines chart maps from p-adics to reals (rather than p-adics!) and vice versa and preferred extremal property allows to complete the discrete image to hopefully unique space-time surface so that topological and algebraic approach are combined. Without preferred extremal property one can complete to smooth real manifold (say) but the completion is much less unique - which indeed conforms with finite pinary resolution.
  3. Also the notion of integration can be defined. If the integral for - say- real curve at the map leaf exists, its value on the p-adic side for its pre-image can be defined by algebraic continuation in the case that it exists. Therefore one can speak about lengths, volumes, action integrals, and similar things in p-adic context. One can also generalize the notion of differential form and its holomomorphic variant and their integrals to the p-adic context. These generalizations allow a generalization of integral calculus required by TGD and also provide a justification for some basic assumptions of p-adic mass calculations.

Could path connectedness have quantal description?

The physical content of path connectedness might also allow a formulation as a quantum physical rather than primarily topological notion, and could boil down to the non-triviality of correlation functions for second quantized induced spinor fields essential for the formulation of WCW spinor structure. Fermion fields and their n-point functions could become part of a number theoretically universal definition of manifold in accordance with the TGD inspired vision that WCW geometry - and perhaps even space-time geometry - allow a formulation in terms of fermions.

The natural question of physicist is whether quantum theory could provide a fresh number theoretically universal approach to the problem. The basic underlying vision in TGD framework is that second quantized fermion fields might allow to formulate the geometry of "world of classical worlds" (WCW) (for instance, Kähler action for preferred extremals and thus Kähler geometry of WCW would reduce to Dirac determinant. Maybe even the geometry of space-time surfaces could be expressed in terms of fermionic correlation functions.

This inspires the idea that second quantized fermionic fields replace the K-valued (K is algebraic extension of p-adic numbers) functions defined on p-adic disk in the construction of Berkovich. The ultrametric norm for the functions defined in p-adic disk would be replaced by the fermionic correlation functions and different Berkovich norms correspond to different measurement resolutions so that one obtains also a connection with hyper-finite factors of type II1. The existence of non-trivial fermionic correlation functions would be the counterpart for the path connectedness at space-time level. The 3-surfaces defining boundaries of a connected preferred extremal are also in a natural manner "path connected": the "path" is defined by the 4-surface. At the level of WCW and in zero energy ontology (ZEO) WCW spinor fields are analogous to correlation functions having collections of these disjoint 3-surfaces as arguments. There would be no need to complete p-adic topology to a path connected topology in this approach.

It must be emphasized that this apporach should be consistent with the first option and that it is much more speculative that the first option.

About literature

It is not easy to find readable literature from these topics. The Wikipedia article about Berkovich space is written with a jargon giving no idea about what is involved. There are video lectures about Berkovich spaces. The web article about Berkovich spaces by Temkin seems too technical for a non-specialist. The slides however give a concise bird's eye of view about the basic idea behind Berkovich spaces.

Topics of the article

The article was originally meant to discuss p-adic icosahedron. Although the focus was redirected to the notion of p-adic manifold - especially in TGD framework - I decided to keep the original starting point since it provides a concrete manner to end up with the deep problems of p-adic manifold theory and illustrates the group theoretical ideas.

  • In the first section icosahedron is described in real context. In the second section the ideas related to its generalization to the p-adic context are introduced. After that I discuss how to define sphere in p-adic context.
  • In the section about algebraic universality I consider the problems related to the challenge of defining p-adic manifolds TGD point of view, which is algebraic and involves the fusion of various number fields and number theoretical universality as additional elements.
  • The key section of the article describes the construction of p-adic space-time topology relying on chart maps of p-adic preferred extremals defined by canonical identification in finite measurement resolution and on the completion of discrete chart maps to real preferred extremals of Kähler action. The needed path-connected topology is the topology induced by canonical identification defining real chart maps for p-adic space-time surface. Canonical identification allows also the definition of p-adic valued integrals and definition of p-adic differential forms crucial in quqantum TGD.
  • Last section discusses in rather speculative spirit the possibility of defining space-time surfaces in terms of correlation functions of induced fermion fields.

For details see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.



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